Asymptotic-numerical study of supersensitivity for generalized Burgers equations

This article addresses some asymptotic and numerical issues related to the solution of Burgers' equation, $-\epsilon u_{xx} + u_t + u u_x = 0$ on $(-1,1)$, subject to the boundary conditions $u(-1) = 1 + \delta$, $u(1) = -1$, and its generalization to two dimensions, $-\epsilon \Delta u + u_t + u u_x + u u_y = 0$ on $(-1,1) \times (-\pi, \pi)$, subject to the boundary conditions $u|_{x=1} = 1 + \delta$, $u|_{x=-1} = -1$, with $2\pi$ periodicity in $y$. The perturbation parameters $\delta$ and $\epsilon$ are arbitrarily small positive and independent; when they approach 0, they satisfy the asymptotic order relation $\delta = O_s ({\rm e}^{-a/\epsilon})$ for some constant $a \in (0,1)$. The solutions of these convection-dominated viscous conservation laws exhibit a transition layer in the interior of the domain, whose position as $t\to\infty$ is supersensitive to the boundary perturbation. Algorithms are presented for the computation of the position of the transition layer at steady state. The algorithms generalize to viscous conservation laws with a convex nonlinearity and are scalable in a parallel computing environment.Comment: 18 pages, 9 tables, 4 figures. Submitted to SIAM J. Scientific Computin

The Regionally-Implicit Discontinuous Galerkin Method: Improving the Stability of DG-FEM

Discontinuous Galerkin (DG) methods for hyperbolic partial differential equations (PDEs) with explicit time-stepping schemes, such as strong stability-preserving Runge-Kutta (SSP-RK), suffer from time-step restrictions that are significantly worse than what a simple Courant-Friedrichs-Lewy (CFL) argument requires. In particular, the maximum stable time-step scales inversely with the highest degree in the DG polynomial approximation space and becomes progressively smaller with each added spatial dimension. In this work we introduce a novel approach that we have dubbed the regionally implicit discontinuous Galerkin (RIDG) method to overcome these small time-step restrictions. The RIDG method is based on an extension of the Lax-Wendroff DG (LxW-DG) method, which previously had been shown to be equivalent to a predictor-corrector approach, where the predictor is a locally implicit spacetime method (i.e., the predictor is something like a block-Jacobi update for a fully implicit spacetime DG method). The corrector is an explicit method that uses the spacetime reconstructed solution from the predictor step. In this work we modify the predictor to include not just local information, but also neighboring information. With this modification we show that the stability is greatly enhanced; in particular, we show that we are able to remove the polynomial degree dependence of the maximum time-step and show how this extends to multiple spatial dimensions. A semi-analytic von Neumann analysis is presented to theoretically justify the stability claims. Convergence and efficiency studies for linear and nonlinear problems in multiple dimensions are accomplished using a MATLAB code that can be freely downloaded.Comment: 26 pages, 4 figures, 8 table

High Order Implicit-Explicit General Linear Methods with Optimized Stability Regions

In the numerical solution of partial differential equations using a method-of-lines approach, the availability of high order spatial discretization schemes motivates the development of sophisticated high order time integration methods. For multiphysics problems with both stiff and non-stiff terms implicit-explicit (IMEX) time stepping methods attempt to combine the lower cost advantage of explicit schemes with the favorable stability properties of implicit schemes. Existing high order IMEX Runge Kutta or linear multistep methods, however, suffer from accuracy or stability reduction. This work shows that IMEX general linear methods (GLMs) are competitive alternatives to classic IMEX schemes for large problems arising in practice. High order IMEX-GLMs are constructed in the framework developed by the authors [34]. The stability regions of the new schemes are optimized numerically. The resulting IMEX-GLMs have similar stability properties as IMEX Runge-Kutta methods, but they do not suffer from order reduction, and are superior in terms of accuracy and efficiency. Numerical experiments with two and three dimensional test problems illustrate the potential of the new schemes to speed up complex applications

A linearly implicit conservative difference scheme for the generalized Rosenau-Kawahara-RLW equation

This paper concerns the numerical study for the generalized Rosenau-Kawahara-RLW equation obtained by coupling the generalized Rosenau-RLW equation and the generalized Rosenau-Kawahara equation. We first derive the energy conservation law of the equation, and then develop a three-level linearly implicit difference scheme for solving the equation. We prove that the proposed scheme is energy-conserved, unconditionally stable and second-order accurate both in time and space variables. Finally, numerical experiments are carried out to confirm the energy conservation, the convergence rates of the scheme and effectiveness for long-time simulation.Comment: accepted in Applied Mathmatics and Computation

Resolve subgrid microscale interactions to discretise stochastic partial differential equations

Constructing discrete models of stochastic partial differential equations is very delicate. Stochastic centre manifold theory provides novel support for coarse grained, macroscale, spatial discretisations of nonlinear stochastic partial differential or difference equations such as the example of the stochastically forced Burgers' equation. Dividing the physical domain into finite length overlapping elements empowers the approach to resolve fully coupled dynamical interactions between neighbouring elements. The crucial aspect of this approach is that the underlying theory organises the resolution of the vast multitude of subgrid microscale noise processes interacting via the nonlinear dynamics within and between neighbouring elements. Noise processes with coarse structure across a finite element are the most significant noises for the discrete model. Their influence also diffuses away to weakly correlate the noise in the spatial discretisation. Nonlinear interactions have two further consequences: additive forcing generates multiplicative noise in the discretisation; and effectively new noise processes appear in the macroscale discretisation. The techniques and theory developed here may be applied to soundly discretise many dissipative stochastic partial differential and difference equations.Comment: Revise

A Goal-Oriented Adaptive Discrete Empirical Interpolation Method

In this study we propose a-posteriori error estimation results to approximate the precision loss in quantities of interests computed using reduced order models. To generate the surrogate models we employ Proper Orthogonal Decomposition and Discrete Empirical Interpolation Method. First order expansions of the components of the quantity of interest obtained as the product between the components gradient and model residuals are summed up to generate the error estimation result. Efficient versions are derived for explicit and implicit Euler schemes and require only one reduced forward and adjoint models and high-fidelity model residuals estimation. Then we derive an adaptive DEIM algorithm to enhance the accuracy of these quantities of interests. The adaptive DEIM algorithm uses dual weighted residuals singular vectors in combination with the non-linear term basis. Both the a-posteriori error estimation results and the adaptive DEIM algorithm were assessed using the 1D-Burgers and Shallow Water Equation models and the numerical experiments shows very good agreement with the theoretical results.Comment: 34 pages, 19 figure

A New Family of Weighted One-Parameter Flux Reconstruction Schemes

The flux reconstruction (FR) approach offers a flexible framework for describing a range of high-order numerical schemes; including nodal discontinuous Galerkin and spectral difference schemes. This is accomplished through the use of so-called correction functions. In this study we employ a weighted Sobolev norm to define a new extended family of FR correction functions, the stability of which is affirmed through Fourier analysis. Several of the schemes within this family are found to exhibit reduced dissipation and dispersion overshoot. Moreover, many of the new schemes possess higher CFL limits whilst maintaining the expected rate of convergence. Numerical experiments with homogeneous linear convection and Burgers turbulence are undertaken, and the results observed to be in agreement with the theoretical findings

On adaptive timestepping for weakly instationary solutions of hyperbolic conservation laws via adjoint error control

We study a recent timestep adaptation technique for hyperbolic conservation laws. The key tool is a space-time splitting of adjoint error representations for target functionals due to S\"uli and Hartmann. It provides an efficient choice of timesteps for implicit computations of weakly instationary flows. The timestep will be very large in regions of stationary flow, and become small when a perturbation enters the flow field. Besides using adjoint techniques which are already well-established, we also add a new ingredient which simplifies the computation of the dual problem. Due to Galerkin orthogonality, the dual solution {\phi} does not enter the error representation as such. Instead, the relevant term is the difference of the dual solution and its projection to the finite element space, {\phi}-{\phi}h . We can show that it is therefore sufficient to compute the spatial gradient of the dual solution, $w = {\nabla} {\phi}$. This gradient satisfies a conservation law instead of a transport equation, and it can therefore be computed with the same algorithm as the forward problem, and in the same finite element space. We demonstrate the capabilities of the approach for a weakly instationary test problem for scalar conservation laws

Explicit and Implicit Kinetic Streamlined-Upwind Petrov Galerkin Method for Hyperbolic Partial Differential Equations

A novel explicit and implicit Kinetic Streamlined-Upwind Petrov Galerkin (KSUPG) scheme is presented for hyperbolic equations such as Burgers equation and compressible Euler equations. The proposed scheme performs better than the original SUPG stabilized method in multi-dimensions. To demonstrate the numerical accuracy of the scheme, various numerical experiments have been carried out for 1D and 2D Burgers equation as well as for 1D and 2D Euler equations using Q4 and T3 elements. Furthermore, spectral stability analysis is done for the explicit 2D formulation. Finally, a comparison is made between explicit and implicit versions of the KSUPG scheme.Comment: 30 pages, 22 figure

Symmetry-preserving finite element schemes: An introductory investigation

Using the method of equivariant moving frames, we present a procedure for constructing symmetry-preserving finite element methods for second-order ordinary differential equations. Using the method of lines, we then indicate how our constructions can be extended to (1+1)-dimensional evolutionary partial differential equations, using Burgers' equation as an example. Numerical simulations verify that the symmetry-preserving finite element schemes constructed converge at the expected rate and that these schemes can yield better results than their non-invariant finite element counterparts.Comment: 21 pages, 3 figure